A category consists of a collection of objects together with morphisms between these objects. Thus, naively, we may think of objects as the ‘elements’ of a category.
More generally, in higher category theory the objects of an -category are the -dimensional cells of that structure, the -morphisms.
If a set is regarded as a discrete category (with no nontrivial morphisms) then the objects of that category are precisely the elements of the set.
In the fundamental groupoid of a topological space , the objects are the points of .
In the category Set, the objects are sets; in Vect the objects are vector spaces; in Top the objects are topological spaces, etc.
If a simplicial set that is a Kan complex is regarded as an -groupoid, then its vertices are the objects of that -groupoid.
Similarly if a simplicial set that is a quasi-category is regarded as an -category, then its vertices are the objects of that -category.
If a globular set is equipped with the structure of a strict ω-category, then its -cells are the objects of that ω-category.
Revised on October 17, 2011 21:48:38
by Toby Bartels